1. Antiderivatives and Indefinite Integration
Lesson 5.1

2. Reversing Differentiation
An antiderivative of function f is
a function F
which satisfies F’ = f
Consider the following:
We note that two antiderivatives of the same function differ by a constant

3. Reversing Differentiation
General antiderivatives f(x) = 6x F(x) = 2x3 + C
because F’(x) = 6x2
k(x) = sec2(x) K(x) = tan(x) + C
because K’(x) = k(x)

4. Differential Equation
A differential equation in x and y involves
x, y, and derivatives of y
Examples
Solution – find a function whose derivative is the differential given

5. Differential Equation
When
Then one such function is
The general solution is

6. Notation for Antiderivatives
We are starting with
Change to differential form
Then the notation for antiderivatives is
"The antiderivative of f with respect to x"

7. Basic Integration Rules
Note the inverse nature of integration and differentiation
Note basic rules, pg 286

8. Practice
Try these

9. Finding a Particular Solution
Given
Find the specific equation from the family of antiderivatives, which contains the point (3,2)
Hint: find the general antiderivative, use the given point to find the value for C

10. Assignment A
Lesson 5.1 A
Page 291
Exercises 1 – 55 odd

11. Slope Fields Slope of a function f(x) Suppose we know f ‘(x) Example
at a point a
given by f ‘(a)
Suppose we know f ‘(x)
substitute different values for a
draw short slope lines for successive values of y
Example

12. Slope Fields For a large portion of the graph, when
We can trace the line for a specific F(x)
specifically when the C = -3

13. Finding an Antiderivative Using a Slope Field
Given
We can trace the version of the original F(x) which goes through the origin.

14. Vertical Motion
Consider the fact that the acceleration due to gravity a(t) = -32 fps2
Then v(t) = -32t + v0
Also s(t) = -16t2 + v0t + s0
A balloon, rising vertically with velocity = 8 releases a sandbag at the instant it is 64 feet above the ground
How long until the sandbag hits the ground
What is its velocity when this happens?
Why?

15. Note Spreadsheet Example
Rectilinear Motion
A particle, initially at rest, moves along the x-axis at acceleration
At time t = 0, its position is x = 3
Find the velocity and position functions for the particle
Find all values of t for which the particle is at rest
Note Spreadsheet Example

16. Assignment B
Lesson 5.1 B
Page 292
Exercises 57 – 93, EOO